aequationes mathematicae

, Volume 39, Issue 2–3, pp 149–160 | Cite as

Trivialization of fans in planar ternary rings with rational prime field

  • Franz B. Kalhoff
Research Papers


Making use of the intimate relations between real places and orderings, we continue studying the spaces of orderings of planar ternary rings (PTRs) with rational prime field. As in the classical case, given a preorderingS of such a PTRT, then the productA S of all natural place ringsA p associated to the orderingsP ∈ X/S is itself a place ring ofT. In particular, ifS is a nontrivial fan ofT thenA s ≠ T. Thus L. Bröcker's celebrated theorem on the trivialization of fans also applies to our setting:

For any fan Sof a PTRT with rational prime field there exists a place µ: T → T′ ∪ {∞} such that the push down S′:= μ(S)\{0,∞} of S is a trivial fan of T′.

Further, Bröcker's global stability formula and, by means of an approximation theorem, some classical, valuation theoretic characterizations of SAP-preorderings and fans also extend to PTRs with rational prime field.

AMS (1980) subject classification

Primary 51G05 Secondary 11E81, 12K05, 12K10 


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Copyright information

© Birkhäuser Verlag 1990

Authors and Affiliations

  • Franz B. Kalhoff
    • 1
  1. 1.Fachbereich MathematikUniversität DortmundDortmund 50West Germany

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